Hi, I lived in this area for 10 years, and moved away almost 10 years ago, but I'll do my best to help.
One apartment community I remember near the Cedar/Warrensville area was called Huntington Green. It is on Cedar Rd in between Mirimar and Green Rds. There was a high rise building and some garden apartments that my friends lived in and seemed pretty nice. Here is their website:
Goldberg Companies, Inc / GCI Residential
At the Warrensville/Fairmount circle in front of JCU, the two story brick w/ white trim apartments were supposed to be nice, and I knew some teachers who lived there and liked it. I think this is their site, but not sure.
Fairmount Circle Apartments University Heights, OH - Apartments for Rent in University Heights Ohio
There are two apartment buildings (2 story brick) at the corner of Cedar/Mirimar, diagonal from Burger King...avoid those. Old, not kept up and full of John Carroll U kids...I used to live there and it was terrible. They look similar to the Huntington Green garden apartments but have a separate owner.
You mentioned Severance Center...if that is near the Walmart, 10 years ago it was a very scary and terrible place to be at night. Check this out to see what the crime is now.
Don't know much about Beachwood except the shopping is even better than it was when I lived near there! I would think they have newer places than Univ Hts.
Good luck, and let me know if you have any other questions!